So if I were to leave this question alone for a while you'd probably get a lot of answers about how the Roman people were tired of a century of civil war and how they decided that an absolute ruler who at least made a show of being a supporter of Republicanism was a fair price to pay for peace. Which is sort of true, but not really. That's a very antiquated view, from before Syme's work on Augustus right before WWII, and it's an idea that's starting to be revived unfortunately. The reality is that they had no choice.

First of all, there's a lot that's different between Caesar and Augustus. Caesar wanted to be number one, sure, but he was fine with leaving his rivals alone if they accepted that he was in charge. Octavian's career was spent brutally annihilating anyone who dared challenge him. Furthermore, Caesar was a great deal less brutal than Octavian had been. Caesar defeated his enemies in battle and forced them out of the senate, but Octavian? Octavian massacred a force of dissenters at Perusia, wiped out Sextus Pompey's troops, crushed Lepidus' aspirations, and annihilated much of northern Italy. Not to mention the fact that his proscriptions were by far the largest proscriptions in Roman history, much larger than Sulla's famed proscriptions. I wrote a writeup here that more or less answers your question, but I'll add to it as well. See, Octavian controlled the entire army (and all its veterans, scattered throughout settlements he had given them in Italy and undoubtedly ready to rise up if anyone spoke out against him), and after the War of Actium he personally controlled the entire east as well, since most of the provinces and cities were forced to swear personal loyalty to him. Furthermore, Octavian personally controlled all of Caesar's old conquests, and he controlled Italy as well. How did he control Italy? Well apart from the proscriptions and the outright massacres of his enemies in Italy, his actual legal power over Italy began shortly after he defeated Sextus Pompey and exiled Lepidus, when the senate (under duress) granted him tribunate sacrosanctity. That meant that he, like the tribune of the plebs, could not be killed and it granted him an enormous amount of power, since he could claim to be acting as a tribune (even though he wasn't actually a tribune). This alone made it difficult for the senate or magistrates to speak against him, along with the fact that after Caesar's wars there were few people left in the senate who were not Caesarians or rather despicable upper-class thugs (which are not mutually exclusive--Caesar had a difficult time policing his supporters, many of which were rather wealthy lowlives), and after Antony's defeat of the last Republicans and the proscriptions and massacres of political enemies there were even fewer left in the senate who weren't complete supporters of Octavian and Antony, or just cowards. Octavian used his political influence gained from his defeat of Pompey and from his new powers to cleanse the senate, essentially forcing many of them to leave or go over to Antony. But what really nailed the coffin was Octavian's actions just before the War of Actium. Before setting sail for Greece Octavian, with the help of the army, forced every community in Italy to swear him an oath of loyalty. Mind you, not to swear an oath to Rome, or the state, or anything like that. A personal oath of loyalty to Octavian alone. This was unprecedented, and forced through purely by force of arms and by the fact that there was no one in the west left to stop him (and to be honest it's debatable whether Antony at this stage had any chance at all--in any case his victory would have resulted in his replacing Octavian as supreme tyrant, nothing more). He continued on, however, by surrounding the Curia with his praetorians and ordering the senate to swear a similar oath of loyalty to himself and to grant him extraordinary powers. Who was going to stop him? It's primarily because of his brutality early in his career that afterwards Augustus went to such great lengths to establish a front of legitimacy (going so far as to commission poetry and great works of art, rewrite the laws to emulate the stuffy stoic laws of the earlier part of the Republic, pretending to be just a private citizen, inventing the First Triumvirate, and blowing the threat to Rome by Antony and Cleopatra into a much bigger deal than it probably was)

I'd like to add a short remark on top of the above answer because I feel that your question is grounded in a misunderstanding of the situation. When Octavian won at Actium in 31 BCE he was immensely powerful. Yet, it took 4 years before he was offered the title of Augustus, why? Well, while /u/XenophonTheAthenian calls Octavian's way of waging power "brutality", I call it cunningness. He realised that by giving up the power at the very day the civil war was over would just lead to more riots and uprisings. Instead he stablized the empire before graciously (if you will) attempting to hand power back to the senate.

This is seen as one of the greater moves Octavian made during his life, simply because it implied that he viewed the senate and the republic as greater than himself. The senate presumably realised the absurdity of the situation, but nonetheless appreciated the move and in return offered Octavian the title Augustus. Augustus repeated this ritual several times during the next decade.

So in essence, the senate only offered him this absolute power because he already had it, they didn't give up anything they had but through the simple act of backing Octavian, the senate remained a power in Rome - at least ostensibly.